Write $(((a\times b)\times a)\times b)\times a\cdot b$, where $a\cdot b = p$, $a\cdot a = q$, and $b\cdot b = r$ in terms of only dot products.
EDIT:
The answer was $(((a\times b)\times a)\times b)\times a\cdot b=((qb−pa)\times b)\times a\cdot b=0−p(a\times b)\times a\cdot b=−p(qb−pa)\cdot b=−p(qr−p2)$, unfortunately i didnt get it on time, but thanks for the help
Recall the triple vector product $$ (a\times b)\times c=(c\cdot a)b-(c\cdot b)a $$ and proceed to simplify the given expression: \begin{align*} [(((a\times b)\times a)\times b), a, b] &=[((a\cdot a)b-(a\cdot b)a)\times b), a, b]\\ &=\dots \end{align*}